Systems and methods for designing zero-shift supercell halftone screens

ABSTRACT

Conventional design tools were not developed for designing square zero-shift supercells. Conventionally, solutions that enable square zero-shift supercells were found by trial and error or by exhaustive analysis. According to a first design criterion of this invention, a non-square supercell in a first frame of reference has a diagonal that is equal in length to the diagonal of a square supercell in a second frame of reference rotated at a desired screen angle to the first frame of reference. The screen angle is a function of the lengths of the sides of the non-square supercell in the first frame of reference. According to a second design criterion, if the area of the corresponding square supercell in the second frame of reference is an integer, a square zero-shift supercell can be designed based on the lengths of the sides of the non-square supercell in the first frame of reference.

BACKGROUND OF THE INVENTION

[0001] 1. Field of Invention

[0002] This invention relates to designing zero-shift supercell halftonescreens.

[0003] 2. Description of Related Art

[0004] With the advent of inexpensive digital printers, methods andsystems for digital halftoning have become increasingly important. It iswell understood that most digital printers operate in a binary mode,i.e., printing or not printing a halftone dot at a specified location orpixel. Digital halftoning controls the printing of halftone dots, wherespatially averaging of the printed dots provides the illusion of thecontinuous tones present in an original image.

[0005] The most common halftone technique is threshold screening, whichcompares the image value of each pixel in the original image with one ofseveral predetermined threshold levels that are stored in a halftonescreen. If the image value is “darker” than the applied thresholdhalftone level, a spot of ink or toner is printed at that pixel.Otherwise, the pixel is left unprinted, so that the background color ofthe image receiving medium is visible. It is well understood in the artthat the distribution of printed pixels depends on the design of thehalftone screen.

[0006] Halftone screens are typically two-dimensional threshold arraysand are relatively small in comparison to the overall image or documentto be printed. Therefore, the screening process uses an identicalhalftone screen cell repeated for each color separation in a mannersimilar to tiling. The output of the screening process, using asingle-cell halftone dot, includes a binary pattern of multiple small“dots”, which are regularly spaced and are determined by the size andthe shape of the halftone screen. In other words, the screening output,as a two-dimensionally repeated pattern, possesses two fundamentalspatial frequencies, which are completely defined by the geometry of thehalftone screen.

[0007] It should be appreciated that, in the halftoning arts, squarehalftone cells, tiled in a zero-shift manner, can be easily combined toform a supercell. In contrast, non-zero-shift tiling results in abrick-like pattern, where the cells of one row are laterally offsetrelative to the upper and lower adjacent rows. Zero-shift refers to thecorners of each of the square halftone supercells meeting at a commonpoint. FIG. 1 illustrates two halftone supercells that have a non-zeroshift. Because supercells are formed by combining a number of halftonecells, supercells can be used to form a “macro” halftone screen forhalftoning the original image. Because a supercell is, by definition,larger than the individual halftone cells used to form the supercell,the resulting screen can have more threshold levels and can achievebetter visual angles, on average, than the simple cell halftone.Reducing the number of centers in supercells that achieve the desiredeffects increases the efficiency of the supercell in conservingresources such as, for example, memory, processing power, and the like.

[0008] Conventionally, halftone screen designers have a number ofconventional design tools usable to create a halftone screen utilizingsupercells. In general, these conventional tools allow the halftonescreen designer to create supercells based on magnifying Holladay dots.Holladay dots are described in “An Optimal Algorithm For HalftoneGeneration For Display And Hard Copies”, T. Holladay, Proceedings of theSociety for Information Display, Vol. 21, No. 2, pages 185-192, 1980. Asshown in FIG. 1, these conventional Holladay dots are described as athreshold array in an implementation rectangle that includes a shiftbetween rows of tiled rectangles.

[0009] Current PostScript hafltoning implementations have difficultyusing arbitrary Holladay dots. These software implementations of thePostScript standard are usually optimized for PostScript type 3 dots. Inparticular, PostScript type 3 dots are zero-shift-square tiles that abutat the corners, as outlined above. These software implementations of thePostScript standard also work most efficiently when these square tilescontain a multiple of 32 pixels per tile.

SUMMARY OF THE INVENTION

[0010] However, conventional supercell schemes attempt to fit asupercell among halftone screens at specified desired angles. Thisprocess usually involves a priori knowledge of the printer resolution,the angle of the halftone screen relative to the raster of the printer,and the frequency of the halftone cells. Typically, conventionalzero-shift supercell design schemes also require searching all of thepossible sizes for the base halftone cell to arrive at the base halftonecell size that facilitates the placement of the centers of the basehalftone cells at the corners of the square zero-shift supercells.

[0011] Unfortunately, the above-outlined conventional design tools werenot designed with the requirements for square zero-shift supercells inmind. Thus, a potential solution for a particular halftone screenprovided by these design tools would be located without taking intoconsideration the requirements for zero-shift-supercells. Thus,conventionally, a solution that enables square zero-shift supercells wasfound either by trial and error or by exhaustive analysis. Thus,designing halftone screens that use square zero-shift supercells usingthese tools is inherently an inefficient process.

[0012] This invention provides systems and methods for efficientlylocating a zero-shift supercell solution for a desired halftone screen.

[0013] This invention separately provides systems and methods forfinding zero-shift supercell using a rotated frame of reference.

[0014] This invention separately provides systems and methods that allowa zero-shift supercell solution for a desired halftone screen to beobtained based on printer resolution and desired screen frequency.

[0015] This invention separately provides systems and methods that allowzero-shift supercells to be located based on a desired effective visualarea for the base halftone cell.

[0016] This invention separately provides systems and methods forlocating zero-shift-halftone solutions based on a desired screen angle.

[0017] In various exemplary embodiments, the systems and methods of thisinvention make use of a number of design criteria discovered by theinventor of this application. In particular, according to a first designcriterion, a non-square supercell in a first frame of reference has adiagonal that is equal in length to the diagonal of a square supercellin a second frame of reference that is rotated at a desired screen angleto the first frame of reference. In particular, the screen angle betweenthe first and second frames of reference is a function of the lengths ofthe sides of the non-square supercell in the first frame of reference.In various exemplary embodiments, the first frame of reference isaligned with the dots that comprise the halftone screen. In variousexemplary embodiments, the second frame of reference is aligned with theoutput device raster.

[0018] According to a second design criterion, if the area of thecorresponding square supercell in the second frame of reference is aninteger, a square zero-shift supercell can be designed based on thelengths of the sides of the non-square supercell in the first frame ofreference.

[0019] In general, conventional Holladay methods would identify thesquare supercell in the second frame of reference as a potential squarezero-shift supercell only if the effective area of the non-squaresupercell in the first frame of reference were a perfect square.

[0020] In various exemplary embodiments, the systems and methodsaccording to this invention can be used to determine one or more sets ofside lengths for the non-square rectangle in the first frame ofreference based on a desired screen angle between the first and secondframes of reference. Then, based on the determined side lengths, aresolution of the image forming device on which the square zero-shifthalftone screen is to be used and/or the desired screen frequency, anestimated effective visual area of a base halftone cell can bedetermined. From this estimated effective visual area of the basehalftone cell, a side length for the square zero-shift supercell can bedetermined and an actual effective visual area for the resulting basehalftone cell can be determined. An actual screen frequency based on theactual effective visual area can then be determined.

[0021] In various other exemplary embodiments, the systems and methodsof this invention can be used to design the square zero-shift supercellbased on a desired area of the square zero-shift supercell, which mustbe a perfect square of the integer side length of the square zero-shiftsupercell. The area of the zero-shift supercell is a function of theside lengths of the non-square supercell in the first frame of referenceand the actual visual area of each base halftone cell of the zero-shiftsupercell. Alternatively, the actual effective visual area of the basehalftone cells making up the square zero-shift supercell can be selectedsuch that the supercell area is a perfect square. In either case, theactual screen frequency of the resulting square zero-shift supercell isa function of the resolution of the image forming device on which thesquare zero-shift supercell halftone screen will be used and the size ofthe actual effective visual area of the base halftone cells that make upthe square zero-shift supercell.

[0022] These and other features and advantages of this invention aredescribed in, or are apparent from, the following detailed descriptionof various exemplary embodiments of the systems and methods according tothis invention.

BRIEF DESCRIPTION OF THE DRAWINGS

[0023] Various exemplary embodiments of this invention will be describedin detail, with reference to the following figures, wherein:

[0024]FIG. 1 illustrates a pair of 9-center Holladay cells having anon-zero shift;

[0025]FIG. 2 illustrates how redundant copies of the Halladay block areused in creating a square zero-shift supercell according to thisinvention;

[0026]FIG. 3 illustrates uniformly tiled square base halftone cells anda first-non-square supercell in a first frame of reference;

[0027]FIG. 4 illustrates an exemplary zero-shift supercell formed in asecond frame of reference that is rotated relative to the first frame ofreference at an angle related to the diagonal of the non-squaresupercell;

[0028]FIG. 5 illustrates the component vectors that represent the squareas exemplary square and non-square supercells of FIG. 3 and FIG. 4, andthe diagonals of the square and non-square supercells;

[0029]FIG. 6 illustrates how the components of the vectors in the secondframe of reference relate trigonometrically to the vectors in the firstframe of reference;

[0030]FIG. 7 is a flowchart outlining a first exemplary embodiment of amethod for designing a square zero-shift supercell according to thisinvention;

[0031]FIG. 8 is a flowchart outlining a second exemplary embodiment of amethod for designing a square zero-shift supercell according to thisinvention;

[0032]FIG. 9 is a flowchart outlining a third exemplary embodiment of amethod for designing a square zero-shift supercell according to thisinvention;

[0033]FIG. 10 is a flowchart outlining a fourth exemplary embodiment ofa method for designing a square-shift supercell according to thisinvention;

[0034]FIG. 11 is a block diagram of a first exemplary embodiment of asystem for designing a square zero-shift supercell according to thisinvention;

[0035]FIG. 12 is a block diagram of a second exemplary embodiment of asystem for designing a square zero-shift supercell according to thisinvention;

[0036]FIG. 13 is a block diagram of a third exemplary embodiment of asystem for designing a square zero-shift supercell according to thisinvention; and

[0037]FIG. 14 is a block diagram of a fourth exemplary embodiment of asystem for designing a square-shift supercell according to thisinvention.

DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS

[0038] Conventional square zero-shift supercell halftoning schemesrequire at least a search for all possible sizes of the zero-shiftsupercell to locate a square zero-shift supercell that is aligned withthe centers of an integer number of halftone dots on each edge of thesquare zero-shift supercell. It should be appreciated that such a searchresults in a set of candidate square zero-shift supercells. A compromiseis effected between the angle of the candidate square zero-shiftsupercell, the frequency spacing of the square zero-shift supercell, andthe resolution of the printer on which the resulting halftone screenwill be used. These factors result in conventional square zero-shiftsupercells having non-ideal angles and/or non-ideal square zero-shiftsupercell frequencies.

[0039] This invention provides systems and methods for designing ahalftone screen having a square zero-shift supercell. In particular, invarious exemplary embodiments, the systems and methods of this inventionuse a rotated second frame of reference, angled relative to a firstframe of reference, provided in units normalized to dot centerdistances. Accordingly, because the first and second frames of referenceare based on a unit cell, rather than in raster units, the exemplarysupercell design systems and methods do not require a priori knowledgeof the printer resolution.

[0040] As briefly outlined above, one conventional method for designingzero-shift supercells comprises magnifying a base Holladay block. FIG. 1shows a base halftone screen 100 comprising a plurality of square basehalftone cells 110 having centers 112. As shown in FIG. 1, a simple 3,1Holladay dot 114 can be magnified to form a magnified Holladay block120. In particular, as shown in FIG. 1, the magnified Holladay block 120is a 9-center dot. That is, each base halftone cell 110 is considered asa unit cell.

[0041] It can be shown geometrically that the area of the simpleHolladay block 114 is equal to the combined area of one base halftonecell 110, and that the area of the magnified Holladay block 120 is equalto the area of nine base halftone cells 110. This can also be seenintuitively because the corners of the magnified Holladay blocks 120 arecentered, at least partially, on the centers 112 of the base halftonecells 110. The sides of the magnified Holladay blocks 120 pass throughadditional centers 112 of other base halftone blocks 110.

[0042] Based on these criteria, the effective area of the magnifiedHolladay blocks 120 corresponds to the number of centers 112 fullycontained within the magnified Holladay block 120, plus one half thenumber of centers 112 lying on the edges of the magnified Holladay block120, plus one quarter of the number of centers 112 lying on the cornersof the magnified Holladay block 120. Thus, as shown in FIG. 1, each ofthe magnified Holladay blocks 120 fully includes 6 of the centers 112,while 5 of the centers 112 lie on the edges of the magnified Holladayblocks 120 and 2 centers lie on corners of the magnified Holladay blocks120. Thus, 6+5/2+2/4=6+2.5+0.5=9.

[0043] However, making a zero-shift supercell from a PostScriptimplementation of a magnified multi-center Holladay block would, ingeneral, require many redundant copies of the magnified Holladay blocks.In the worst case, to insure that all four corners of the squarezero-shift supercell land on dot centers making such a square zero-shiftsupercell, would require a square array having the same width as theHolladay block 120. This worst case requires the area of the zero-shiftsupercell be equal to n², rather than n, times the area of the basehalftone cells. Consequently, the zero-shift supercell has n², ratherthan n, times the number of threshold values of the base halftone cell.For the particular example of the expanded Holladay block 120 shown inFIG. 1, it can be shown that a zero-shift square supercell of the samewidth with only ten times the height would align correctly on the fourcorners. The exemplary embodiment of the expanded Holladay block 120shown in FIG. 1 is, however, a special case because n=9 in this example,and 9 is a square number.

[0044]FIG. 2 illustrates the non-special case, where n is not a squarenumber, such that redundant copies of the expanded Holladay block 120are needed. As shown in FIG. 2, for most of the Holladay blocks 120, atleast some portion of those Holladay blocks 120 extend outside of thesquare zero-shift supercell 130. In particular, in the exemplaryembodiment shown in FIG. 2, only the top-most Holladay block 120 liesentirely within the square zero-shift supercell 130. For each other setof laterally-adjacent pairs of Holladay blocks 120, the portion of oneof the pair of Holladay blocks 120 that extends beyond the edge of thesquare zero-shift supercell 130 is equal in area to the portion of theother Holladay block 120 that lies within the bounds of the squarezero-shift supercell 130. That is, when viewed another way, for each rowof Holladay blocks 120, the portion of a Holladay block 120 that extendsbeyond the edges of the square zero-shift supercell 130 is equal to theportion of the zero-shift square supercell 130 associated with that rowthat is not also within that Holladay block 120.

[0045] As briefly outlined above, conventional Holladay methods aresignificantly inefficient when attempting to design halftone screenshaving zero-shift supercells. That is, the zero-shift characteristic ispresent in only a small number of the supercells that can be generatedusing conventional Holladay methods. However, conventional Holladaymethods do not have any techniques for selecting or readily identifyingthose supercells that have the zero-shift characteristic.

[0046] In various exemplary embodiments of the systems and methodsaccording to this invention, based on a desired screen angle for ahalftone screen formed using square zero-shift supercells, a secondframe of reference, which is rotated by the desired screen anglerelative to a first frame of reference, is created. In particular, therotated second frame of reference is normalized to the distance betweenthe dot centers, such as the dot centers 112 shown in FIG. 1 and the dotcenters 212 shown in FIGS. 3-6. In particular, in various exemplaryembodiments of the systems and methods according to this invention, thedesired square zero-shift supercell will appear in the rotated secondframe of reference as a square rotated to the first frame of reference,where the corners of the square zero-shift supercell are aligned withthe dot centers of the base halftone cells 110.

[0047]FIG. 3 shows a second halftone screen 200 having a plurality ofsquare uniformly sized base halftone cells 210 having centers 212. Thehalftone screen 200 defines a first frame of reference having an x-axis202 and a y-axis 204. A non-square halftone supercell 220 can be formedin the halftone screen 200 having integer values for the orthogonalsides 222 and 224. In general, the first side 222 will have a length N,while the second side 224 will have a length M, where N≠M. The halftonesupercell 220 will have a diagonal 226 having a length H that is relatedto the lengths N and M of the first and second sides 222 and 224 by thePythagorean theorem, i.e.:

H={square root}{square root over ((N ² +M ²))}.   (1)

[0048]FIG. 4 shows a square supercell 230 aligned to a second frame ofreference defined by the x′ axis 206 and the y′ axis 208 superimposed ata desired screen angle θ over the halftone screen 200 shown in FIG. 3.In particular, as shown in FIG. 4, the square zero-shift supercell 230has two orthogonal sides 232 and 234 having lengths P and Q,respectively. However, since the supercell 230 is, by definition,square, the lengths P and Q of the first and second sides 232 and 234are equivalent. Additionally, and most importantly, the diagonal 226 ofthe non-square halftone supercell 220 that is aligned with the axes 202and 204 of the halftone screen 200 is also the diagonal of the squarezero-shift supercell 230. Again applying the Pythagorean theorem, thelength H of the diagonal 226 in the second frame of reference defined bythe axes 206 and 204 is:

H={square root}{square root over ((P ² +Q ²))}.   (2)

[0049] However, since P and Q are equivalent, as outlined above, Eq. (2)becomes:

H={square root}{square root over ((2P ²))}.   (3)

[0050] Substituting Eq. (3) into Eq. (1):

{square root}{square root over ((2P ²))}={square root}{square root over((N ² +M ²))}.   (4)

[0051] Next, after squaring both sides and solving for P, Eq. (4)becomes:

P ²=(N ² +M ²)/2.   (5)

[0052] Importantly, P² is the area of the square zero-shift supercell230. At the same time, N and M are defined in units of the base halftonecells 210 shown in FIG. 4. That is, M and N are in units of thecenter-to-center distance between two centers 212 of the base halftonecells 210 in the frame of reference of the halftone screen 200 definedby the x and y axes 202 and 204. Thus, the center-to-center distance,when squared, is the area associated with a base halftone cell 210. As aresult, Eq. (5) defines the implementation area of the square zero-shifthalftone cell 230 in units of the number C of dot centers:

C=(N ² +M ²)/2.   (6)

[0053] Thus, as outlined above, many combinations of integer componentsN and M in the first frame of reference can be used to produce anappropriate desired angle θ between the first and second frames ofreference. However, only those combinations where the number C of thedot centers 212 effectively contained within the square zero-shiftsupercell 230, and thus the area of the square zero-shift supercell 230,is an integer number of the areas of the basic halftone cells 210 willresult in an implementable supercell.

[0054]FIG. 5 shows the decomposition of the diagonal 236 of the squarezero-shift supercell 230 that is in the second frame of referencedefined by the x′ and y′ axes 206 and 208 into x axis and y axiscomponents that are aligned with the x and y axes 202 and 204 of thefirst frame of reference. That is, as shown in FIG. 5, the diagonal 236can be decomposed into the orthogonal sides 222 and 224 of a non-squaresupercell that is aligned with the centers 212 of the basic halftonecells 210 and the x and y axes 202 and 204. In the examples shown inFIGS. 3-5, the diagonal 236 of the square zero-shift supercell 230 canbe decomposed into the first and second sides 222 and 224 of thenon-square supercell 230 having side lengths N=4 and M=2, respectively.As a result, according to Eq. (6) the effective number of C of thecenters 212 contained within the square zero-shift supercell 230 is(4²+2²)/2=10. Accordingly, because the effective number C of centers inthe square zero-shift halftone cell 230 is an integer, the cell 230shown in FIGS. 4 and 5 is in fact a square zero-shift supercell.

[0055] Importantly, since 10 is not a perfect square, it would bedifficult to find the square zero-shift supercell 230 having the angle θbetween the first and second frames of reference shown in FIG. 3 usingconventional Holladay methods. It should also be appreciated that, asyet, it is not necessary to assign units to the square zero-shiftsupercell 230 because the angles θ and the number C of centers 212within the square zero-shift supercell 230 can be determined withoutrequiring knowledge of either the resolution or the frequency of theactual halftone screen that will be implemented using this squarezero-shift supercell 230.

[0056] It should also be appreciated that, as shown in FIG. 6, thescreen angle θ between the first and second frames of reference, whichis also the effective visual angle of the halftone screen implemented bythe square zero-shift supercell 230, can be related to the lengths N andM of the first and second sides 222 and 224 of the non-square supercell220. In particular, as shown in FIG. 6, the line h extending between thecenters 212 of two laterally adjacent basic halftone cells 210 can actas the hypotenuse of a right triangle having an interior angle equal toθ, with the other two sides of that triangle aligned with the x′ and y′axes 206 and 208 of the second frame of reference. In this case, theother two sides of this small right triangle will have side lengths pand q, respectively.

[0057] As a result, as shown in FIG. 6, the length P of the first side232 of the implementable square zero-shift supercell 230 can bedetermined by adding the number N of the p sides and subtracting M ofthe q sides, or:

P=(N*p)−(M*q).   (7)

[0058] Similarly, the length of the side 234 of the implementable squarezero-shift supercell 230 can be determined by adding N of the q sidesand M of the p sides, or:

Q=(N*q)+(M*p)   (8)

[0059] Since, by definition, as outlined above, the lengths P and Q ofthe orthogonal sides 232 and 234 of the square zero-shift supercell areequivalent:

(N*p)−(M*q)=(N*q)+(M*p).   (9)

[0060] Solving for p and q:

q/p=(N−M)/(N+M).   (10)

[0061] However, by definition, since p and q are the lengths of thesides of a right triangle having an interior angle θ:

tan(θ)=q/p.   (11)

[0062] Thus, solving for θ:

θ=tan⁻¹((N−M)/(N+M))   (12)

[0063] Alternatively, solving for M:

M=N*(1−tan(θ))/(1+tan(θ)).   (13)

[0064] In particular, Eq. (12) implies that, should the lengths N and Mof the sides 222 and 224 of the non-square supercell 220 be known, thescreen angle θ between the non-square supercell 220 and the squarezero-shift supercell 230 can be determined. Alternatively, Eq. (13)implies that, for a desired screen angle θ between the base halftonecells 210 aligned with the first frame of reference defined by the x andy axes 202 and 204 and the square zero-supercell 230 aligned with thesecond frame of reference defined by the x′ and y′ axes 206 and 208,once an integer value for N is selected, a (probably) non-integer valueM′ can be determined. Then, an actual integer value for M can beselected as an integer close to the non-integer value M′. The selectedvalue N and the determined value M can then be used to determine theactual screen angle θ and number C of the centers according to Eqs. (11)and (6). Of course, it should be appreciated that, in Eq. (13), insteadof solving for M, Eq. (13) could have been developed by solving for N.In this case, for a desired value for the screen angle θ and a selectedvalue for M, a (probably) non-integer value N′ could be determined.Then, an integer value for N could be selected as an integer close tothe non-integer value N′.

[0065] It should also be appreciated that once the angle and the numberof centers is determined, using either Eqs. (6) and (12), or Eqs. (6)and (13), the resolution and frequency can then be considered. Inparticular, the approximate size of the implementable squarezero-supercell 230 can be determined by first estimating the effectivevisual area A_(v) of a single one of the base halftone cells 210 fromthe resolution R of the printer on which the square zero-shift halftonescreen will be implemented and the desired frequency f of that halftonescreen. In particular, the effective visual area A_(v) is:

A _(v)=(R/f)²,   (14)

[0066] where:

[0067] R is the resolution in pixels per inch of the printer on whichthe halftone screen is to be implemented; and

[0068] f is the frequency of that halftone screen in base halftone cells210 per inch.

[0069] Then, the total supercell area A_(s) will be:

A_(s)=A_(v) * C.   (15)

[0070] It should be appreciated, as outlined above, C is the number ofcenters within the square zero-shift halftone cell 230. In particular,the length P of the sides 232 and 234 of the implementable squarezero-supercell 230 will be an integer value that is close to the valueP′, where:

P′={square root}{square root over ((A _(s)))}  (16)

[0071] If the resolution R is 600 pixels per inch and the frequency ffor the halftone screen is desirably close to 150 cells per inch, thenthe actual effective visual area A_(v) is equal to (600/150)², or 16.For the exemplary implementable square zero-shift supercell 230 shown inFIGS. 3-6, C is 10. Thus, the total supercell area A_(s) is 16*10 or160. The approximate side length P′ is thus (160)⁵ or 12.6. The nearestinteger value to 12.6 is 13. Thus, the side length P of theimplementable square shift-supercell 230 is 13. The actual supercellarea A_(s) is thus 132 or 169. Accordingly, the actual effective visualarea A_(v) is A_(s)/C, or 169/10 or 16.9.

[0072] From Eq. (14), and rewriting to solve for the actual frequency f,the actual frequency f is 600/(16.9)^(0.5) or 145.95 dots/inch.

[0073] It should be appreciated that, in general, the effective visualarea A_(v) will not be an integer. In this general case, where theeffective visual area A_(v) is not an integer, in various exemplaryembodiments, the realizable supercell often will be designed withnon-congruent shapes. That is, in various exemplary embodiments,adjacent dot centers within the supercell will not grow identically inshape from level to level. In that case, the angle and/or the frequencyof the dot centers would be exact only on average across the entiresupercell.

[0074] Alternatively, the effective visual area A_(v) could be selectedto be an integer. In this case, there is a good chance that theimplementable square zero-shift supercell 230 can be designed withcongruent centers. For example, if the effective visual area A_(v) isselected as 10, then the implementable square zero-shift supercell 230can be made up of 10 congruent copies of the simple 3,1 Holladay dot 114with 10 pixels each and having an angle θ with a value of 18.43 degrees.In particular, it should be appreciated that the simple 3,1 Holladayblock in this case would have a width of 10 pixels and a height 7 of onepixel. The supercell can be designed with ten identical sub-cells withidentical growth sequences and exact angles and frequencies between dotcenters.

[0075] By definition in this example, the effective visual area A_(v) isselected to be 10, and, M and N are 4 and 2, respectively. Thus, C is 10and the supercell area A_(s) is (10*10) or 100. The length P of the sideof the implementable square zero-shift supercell 230 is (100)⁵, or 10.Again solving Eq. (14) for the frequency f, the actual frequency f is600/(10)⁵ or 189.7 dots per inch.

[0076]FIG. 7 is a flowchart outlining a first exemplary embodiment of amethod of designing a square-shift supercell according to thisinvention. As shown in FIG. 7, beginning in step S100, operationcontinues to step S105, where a desired screen angle θ between the firstand second frames of reference is selected. Next, in step S110, adesired value for either the first side length N or the second sidelength M of the non-square supercell in the first frame of reference isselected. Then, in step S115, the value for the nominal side length M′or N′ is determined based on the selected desired screen angle θ and theselected first or second side length N or M. Operation then continues tostep S120.

[0077] In step S120, the actual side length M or N is selected ordetermined based on the nominal side length M′ or N′ such that bothlengths, as well as the number C of centers in the square zero-shiftsupercell, will all be integer values. Next, in step S125, the actualvalue for the number C of the centers within the square zero-shiftsupercell is determined based on the side lengths M and N. Then, in stepS130, the effective visual area A_(v) of the base halftone cell of thehalftone screen being designed is estimated based on the printerresolution and the desired screen frequency. Operation then continues tostep S135.

[0078] In step S135, the actual supercell area A_(s) is determined basedon the estimated effective visual area A_(v) and the number C of thecenters that are within the square zero-shift supercell. Then, in stepS140, the nominal side length P′ of the square zero-shift supercell isdetermined based on the determined actual supercell area A_(s). Next, instep S145, the actual integer-value side length P is determined based onthe nominal side length P′. Operation then continues to step S150.

[0079] In step S150, the actual effective visual area A_(v) isdetermined based on the actual integer-value side length P. Next, instep S155, the actual screen frequency f is determined based on theactual effective visual area A_(v) and the printer resolution R. Then,in step S160, the method stops.

[0080]FIG. 8 is a flowchart outlining a second exemplary embodiment of amethod for designing a square zero-shift supercell according to thisinvention. In general, the steps outlined in FIG. 8 are similar to thesteps outlined in FIG. 7. The major difference between the flowchartsoutlined in FIGS. 7 and 8 is the order and specific actions performed insteps S205-S220 relative to steps S105-S125.

[0081] In particular, beginning in step S200, operation continues tostep S205, where a desired value for either the first side length N orthe second side length M is selected. Then, in step S210, the sidelength M or the side length N that was not selected or determined instep S205 is selected such that the number C of centers will be aninteger value. Next, in step S215, the numbers C of centers within thesquare zero-shift supercell is determined based on the first and secondside lengths M and N selected in steps S205 and S210. Operation thencontinues to step S220.

[0082] In step S220, the screen angle θ between the first and secondframes of reference is determined based on the side lengths M and Nselected or determined in steps S205 and S210. Control then continues tostep S225. In particular, steps S225-S255 are identical to stepsS130-S160, respectively, shown in FIG. 6. Thus, steps S225-255 will notbe described in further detail.

[0083]FIG. 9 is a flowchart outlining a third exemplary embodiment ofthe method for designing a square zero-shift supercell according to thisinvention. In particular, steps S305-S325, as shown in FIG. 9, areidentical to steps S105-S125 of FIG. 7, as described above. Thus, nofurther description of these steps will be provided.

[0084] In particular, once the number C of centers within the squarezero-shift supercell is determined in step S325, operation continues tostep S330. In step S330, the actual effective visual area A_(v) of thebase halftone cell of the halftone screen being designed is selected.Next, in steps S335, the actual supercell area A_(s) is determined basedon the selected actual effective visual area A_(v) and the determinednumber C of centers. Then, in step S340, the nominal side length P′ ofthe square zero-shift supercell is determined based on the determinedactual supercell area A_(s). Operation then continues to step S345.

[0085] In step S345, the actual integer-valued side length P isdetermined based on the determined nominal side length P′. Then, in stepS350, the actual screen frequency f is determined based on the selectedeffective visual area A_(v) and the printer resolution R. Then, in stepS355, the method ends.

[0086]FIG. 10 is a flowchart outlining a fourth exemplary embodiment ofa method for designing a square zero-shift supercell according to thisinvention. In particular, steps 405-420, as shown in FIG. 10, areidentical to steps S205-S220 described above with respect to FIG. 8.Thus, no further description of these steps will be provided. At thesame time, steps S425-S450, as shown in FIG. 10, are identical to stepsS330-S355 outlined above with respect to FIG. 9. Thus, no furtherdescription of these steps will be provided.

[0087] FIGS. 11-14 are block diagrams outlining first-fourth exemplaryembodiments of square zero-shift supercell designing systems 300-303,respectively, according to this invention. As shown in FIGS. 11-14, thesquare zero-shift supercell designing systems 300-303 includes one ormore of an input/output interface 310, a controller 320, a memory 330, afirst nominal side length determining circuit, routine or application340, a first actual side length selecting or determining circuit,routine or application 350, a center number determining circuit, routineor application 360, an effective visual area estimating circuit, routineor application 370, a supercell area determining circuit, routine orapplication 380, a second nominal side length determining circuit,routine or application 390, a second actual side length determiningcircuit, routine or application 400, an actual effective visual areadetermining circuit, routine or application 410, and an actual screenfrequency determining circuit, routine or application 420, eachinterconnected by one or more control and/or data busses and/or one ormore application programming interfaces 430. Additionally, one or moredata input devices and/or data output devices 305 are connected to theinput/output interface 310.

[0088] As shown in FIGS. 11-14, each of the square zero-shift supercelldesigning systems 300-303 is, in various exemplary embodiments,implemented on a programmed general purpose computer. However, invarious exemplary embodiments, each of the square zero-shift supercelldesigning systems 300-303 is implemented on a special purpose computer,a programmed microprocessor or microcontroller and peripheral integratedcircuit elements, an ASIC or other integrated circuit, a digital signalprocessor, a hardwired electronic or logic circuit such as a discreteelement circuit, a programmable logic device such as a PLD, PLA, FPGA orPAL, or the like. In general, any device, capable of implementing afinite state machine that is in turn capable of implementing theflowcharts shown in FIGS. 7-10, can be used to implement the squarezero-shift supercell designing system 300.

[0089] The memory 330 shown in FIGS. 11-14 can be implemented using anyappropriate combination of alterable, volatile or non-volatile memory ornon-alterable, or fixed, memory. The alterable memory, whether volatileor non-volatile, can be implemented using any one or more of static ordynamic RAM, a floppy disk and disk drive, a writeable or re-rewriteableoptical disk and disk drive, a hard drive, flash memory or the like.Similarly, the non-alterable or fixed memory can be implemented usingany one or more of ROM, PROM, EPROM, EEPROM, an optical ROM disk, suchas a CD-ROM or DVD-ROM disk, and disk drive or the like.

[0090] It should be understood that each of the circuits, routinesand/or applications shown in FIGS. 11-14 can be implemented as portionsof a suitably programmed general-purpose computer. Alternatively, eachof the circuits, routines and/or applications shown in FIGS. 11-14 canbe implemented as physically distinct hardware circuits within an ASIC,or using a FPGA, a PLD, a PLA or a PAL, or using discrete logic elementsor discrete circuit elements. Moreover, the square zero-shift supercelldesigning system 300 shown in FIGS. 11-14 can be implemented as softwareexecuting on a programmed general purpose computer, a special purposecomputer, a microprocessor or the like. The particular form each of thecircuits, routines and/or application shown in FIGS. 11-14 will take isa design choice and will be obvious and predicable to those skilled inthe art.

[0091] In the first exemplary embodiment of the square zero-shiftsupercell designing system 300 shown in FIG. 11, to create a desiredsupercell, the user inputs, using the one or more data input and/oroutput devices 305, data defining a desired screen angle θ between thefirst and second frames of reference. The user also inputs a desiredvalue for one of the first and second side lengths N or M using the oneor more data input and/or output devices 305. Under control of thecontroller 320, the input/output interface 310 provides this data to thememory 330, which stores this data. Then, the first nominal side lengthdetermining circuit, routine or application 340 determines the value forthe nominal side length M′ or N′ of the other side based on the screenangle θ and the side length N or M input through the data input and/oroutput devices 305 and the input/output interface 310.

[0092] Once the nominal side length M′ or N′ is determined, the nominalside length M′ or N′ can be output under control of the controller 310by the input/output interface 310 to the data input and/or outputdevices 305 to allow the user to select an actual side length M or Nbased on the determined nominal side length M′ or N′. Alternatively, thefirst actual side length selecting or determining circuit, routine orapplication 350 can automatically select or determine the actual sidelength M or N. This selection or determination can use any one of anumber of potential techniques. For example, the integer portion of thenominal side length M′ or N′ determined by the first nominal side lengthdetermining circuit or routine could be used as the actual side length.Alternatively, the nominal side length M′ or N′ could be rounded to thenearest integer using standard mathematical techniques.

[0093] Finally, the actual side length could be selected based on atable stored in the memory 330, such as the table set forth below inTable 1. Table 1 could be implemented as a lookup table, where thevalues for M and N are portions of the address to a memory location.TABLE 1 LENGTH OF FIRST SIDE N 1 2 3 4 5 6 7 8 9 10 LENGTH OF SECONDSIDE M 1  1 x  5 x 13 x 25 x 41 x 2 x  4 x 10 x 20 x 34 x  52 3  5 x  9x 17 x 29 x 45 x 4 x 10 x 16 x 26 x 40 x  58 5 13 x 17 x 25 x 37 x 53 x6 x 20 x 26 x 36 x 50 x  68 7 25 x 29 x 37 x 49 x 65 x 8 x 34 x 40 x 50x 64 x  82 9 41 x 45 x 53 x 65 x 81 x 10 x 52 x 58 x 68 x 82 x 100

[0094] Table 1 indicates, for a given side value M or N, the potentiallengths of the other side N or M that can be selected to provide aninteger number of centers. In particular, as shown in Table 1, if theselected side length N or M is even, the actual side length for theother side N or M must also be even. Likewise, if the selected sidelength M or N is odd, the other side length N or M must be odd as well.This occurs because the sum of the squares of M and N itself must beeven to ensure the number C of centers is an integer. The sum of thesquares will be even if only both squares are even or both squares areodd. Furthermore, each squared number M or N will be even or odd only ifthe side lengths M and N are even or odd, respectively.

[0095] Once the actual side length for the other side M or N isselected, using either the first actual side length selecting ordetermining circuit, routine or application 350 or via an input receivedfrom the user via the data input and/or output devices 305 and theinput/output interface 310, the center number determining circuit,routine or application 360, under control of the controller 320,determines the number C of centers, as outlined above with respect toEq. (6).

[0096] Then, as outlined above with respect to Eq. (14), the estimatedeffective visual area is determined by the effective visual areaestimating circuit, routine or application 370. Next, in accordance withEq. (15), the total supercell area is determined using the supercellarea determining circuit, routine or application 380. The second nominalside length determining circuit, routine or application 390 thendetermines the nominal side length of the square zero-shift supercell inaccordance with Eq. (16).

[0097] The actual side length for the square zero-shift supercell isthen selected or determined by the second actual side length determiningcircuit, routine or application 400 as outlined above with respect tothe first actual side length selecting circuit, routine or application350. Alternatively, under control of the controller 320, the nominalside length is output through the input/output interface 310 to the datainput and/or output devices 305 to allow the user to select the actualside length for the square zero-shift supercell.

[0098] The effective visual area determining circuit, routine orapplication 410 then determines the actual effective visual area, asoutlined above with respect to Eq. (15). The actual screen frequencydetermining circuit, routine or application 420 then determines theactual screen frequency as outlined above with respect to Eq. (14).

[0099] Of course, it should be appreciated that, if the user selects thefirst or second actual side lengths and/or selects the actual sidelength of the square zero-shift supercell, the first actual side lengthselecting circuit, routine or application 350 and the second actual sidelength determining circuit, routine or application 400, respectively,can be omitted from the first exemplary embodiment of the squarezero-shift supercell designing system 300.

[0100]FIG. 12 shows the second exemplary embodiment of the squarezero-shift supercell designing system 301 according to this invention.As shown in FIG. 12, the second exemplary embodiment of the squarezero-shift supercell designing system 301 generally contains the samecircuit, routine or application elements as the first exemplaryembodiment of the square zero-shift supercell designing system 300.However, in the second square zero-shift supercell designing system 301,the first nominal side length determining circuit, routine orapplication 340 is omitted entirely, and the first actual side lengthselection circuit, routine or application 350 can be optionally omittedor included. In addition, the second exemplary embodiment of the squarezero-shift supercell designing system 301 includes an angle determiningcircuit, routine or application 440.

[0101] In particular, except as noted below, the operation of the secondexemplary embodiment of the square zero-shift supercell designing system301 is identical to the operation of the first exemplary embodiment ofthe square zero-shift supercell designing system 300. In particular,after receiving an input through the data input devices 305 defining thedesired value for the first or second side length N or M, the secondexemplary embodiment of the square zero-shift supercell designing system301, like the first exemplary embodiment of the square zero-shiftsupercell designing system 300, either automatically selects ordetermines the second actual side length using the first actual sidelength selecting or determining circuit, routine or application 350, or,by omitting the first actual side length selection circuit, routine orapplication 350, receives a further input via the data input and/oroutput devices 305 defining the other of the side length N or M. Then,the angle determining circuit, routine or application 440 determines thescreen angle according to Eq. (12). Once the two side lengths M and Nand the screen angle θ are defined, the operation of the remainingcircuits, routines and/or application 360-420 occurs as outlined abovewith respect to the first exemplary embodiment of the square zero-shiftsupercell designing system 300.

[0102]FIG. 13 is a block diagram of the third exemplary embodiment ofthe square zero-shift supercell designing system 302 according to thisinvention. As shown in FIG. 12, the third square zero-shift supercelldesigning system 302 is generally identical to the first exemplaryembodiment of the square zero-shift supercell designing system 300,except that the effective visual area estimating circuit, routine orapplication 370 and the effective visual area determining circuit,routine or application 410 are omitted.

[0103] In operation, the square zero-shift supercell designing system302, similarly to the first exemplary embodiment of the squarezero-shift supercell designing system 300, inputs the desired screenangle θ and a first one of the first or second side lengths N or M fromthe user via the data input and/or output devices 305 and theinput/output interface 310. However, in addition to these data items,the third exemplary embodiment of the square zero-shift supercelldesigning system 302 also inputs a selected effective visual area of thebase halftone cell from the user through the one or more data inputand/or output devices 305 and the input/output interface 310.

[0104] Subsequently, the first nominal side length determining circuit,routine or application 340 determines a nominal value for the other sidelength M′ or N′ as outlined above. Then, as outlined above with respectto the first exemplary embodiment of the square zero-shift supercelldesigning system 300, the actual value for the side length of the otherside M or N is either input by the user via the one or more data inputand/or output devices 305 and the input/output interface 310 or isautomatically selected or determined using the actual side lengthselecting or determining circuit, routine or application 350.

[0105] Then, the center number determining circuit, routine orapplication 360 operates as outlined above. In this case, the user hasdirectly supplied a selected value for the effective visual area A_(v).As a result, after the center number determining circuit, routine orapplication 360 determines the number C of centers, the supercell areadetermining circuit, routine or application 380, the second nominal sidelength determining circuit, routine or application 390 and the secondactual side length determining circuit, routine or application 400immediately operated as outlined above with respect to the firstexemplary embodiment of the square zero-shift supercell designing system300 based on the effective visual area value supplied by the user.

[0106] Likewise, because the user has directly supplied the selectedeffective visual area, as outlined above, the actual screen frequencydetermining circuit, routine or application 420 then immediatelydetermines the actual screen frequency, as outlined above with respectto the first exemplary embodiment of the square zero-shift supercelldesigning system 300.

[0107]FIG. 14 is a block diagram outlining the fourth exemplaryembodiment of the square zero-shift supercell designing system 303according to this invention. In general, the fourth exemplary embodimentof the square zero-shift supercell designing system 303 is identical tothe second exemplary embodiment of the square zero-shift supercelldesigning system 301, except that, like the third exemplary embodimentof the square zero-shift supercell designing system 302, the effectivevisual area estimating circuit, routine or application 370 and theeffective visual area determining circuit, routine or application 410are omitted. Thus, in operation, the fourth exemplary embodiment of thesquare zero-shift supercell designing system 303 inputs the selecteddesired value for the first or second side length M or N and then eitherautomatically selects or determines, or alternately inputs, the valuefor the other of the side lengths M or N, as outlined above with respectto the second exemplary embodiment of the square zero-shift supercelldesigning system 301. However, like the third exemplary embodiment ofthe square zero-shift supercell designing system 302, the fourthexemplary embodiment of the square zero-shift supercell designing system303 also inputs the selected effective visual area from the user throughthe one or more data input devices 305 and the input/output interface310.

[0108] Subsequently, the angle determining circuit, routine orapplication 440 operates as outlined above with respect to the secondexemplary embodiment of the square zero-shift supercell designing system301, while the remaining circuits, routines or application 360, 380-400and 420 operate as outlined above with respect to the third exemplaryembodiment of the square zero-shift supercell designing system 302.

[0109] While this invention has been described in conjunction with theexemplary embodiments outlined above, it is evident that manyalternatives, modifications and variations will be apparent to thoseskilled in the art. Accordingly, the exemplary embodiments of theinvention, as set forth above, are intended to be illustrative, notlimiting. Various changes may be made without departing from the spiritand scope of the invention.

What is claimed is:
 1. A method for designing a halftone screen having asquare zero-shift halftone supercell, comprising: selecting a desiredangle between a first frame of reference and a second frame ofreference; selecting a desired length for a first side of a pair ofsides of a non-square supercell in the first frame of reference;determining a value for a nominal side length for a second side of thepair of sides of the non-square supercell in the first frame ofreference; selecting an actual side length for the second side based onthe determined nominal side length of the second side such that a numberof centers within the square zero-shift halftone supercell is aninteger; and determining an integer number of centers within the squarezero-shift halftone supercell based on the side lengths of the first andsecond sides of the non-square supercell.
 2. The method of claim 1,further comprising: estimating an effective visual area of a basichalftone cell of the halftone screen based on a resolution of a printerby which the halftone screen will be printed and a desired screenfrequency; and determining a supercell area based on the estimatedeffective visual area and the determined integer number of centers ofthe square zero-shift halftone supercell.
 3. The method of claim 2,further comprising: determining a nominal side length of the squarezero-shift halftone supercell based on the determined supercell area;and determining an actual integer-valued side length of the squarezero-shift halftone supercell based on the determined nominal sidelength of the square zero-shift halftone supercell.
 4. The method ofclaim 3, further comprising: determining an actual effective visual areaof the basic halftone cell based on the actual integer-valued sidelength of the square zero-shift halftone supercell; and determining anactual screen frequency based on the actual effective visual area of thebasic halftone cell and the printer resolution.
 5. The method of claim1, further comprising: selecting an effective visual area of a basichalftone cell of the halftone screen; determining a supercell area basedon the selected effective visual area and the determined integer numberof centers.
 6. The method of claim 5, further comprising: determining anominal side length of the square zero-shift halftone supercell based onthe determined supercell area; and determining an actual integer-valuedside length of the square zero-shift halftone supercell based on thedetermined nominal side length of the square zero-shift halftonesupercell.
 7. The method of claim 6, further comprising determining anactual screen frequency based on the effective visual area of the basichalftone cell and a resolution of a printer by which the halftone screenwill be printed.
 8. A method for designing a halftone screen having asquare zero-shift halftone supercell, comprising: selecting a desiredlength for a first side of a pair of sides of a non-square supercell;selecting a length for a second side of the pair of sides of thenon-square supercell; determining an integer number of centers withinthe square zero-shift halftone supercell based on the side lengths ofthe first and second sides of the non-square supercell.
 9. The method ofclaim 8, further comprising determining an angle between a first frameof reference in which the non-square supercell lies and a second frameof reference in which the square zero-shift halftone supercell liesbased on the selected lengths of the first and second sides of thenon-square supercell.
 10. The method of claim 8, further comprising:estimating an effective visual area of a basic halftone cell of thehalftone screen based on a resolution of a printer by which the halftonescreen will be printed and a desired screen frequency; and determining asupercell area based on the estimated effective visual area and thedetermined integer number of centers of the square zero-shift halftonesupercell.
 11. The method of claim 10, further comprising: determining anominal side length of the square zero-shift halftone supercell based onthe determined supercell area; and determining an actual integer-valuedside length of the square zero-shift halftone supercell based on thedetermined nominal side length of the square zero-shift halftonesupercell.
 12. The method of claim 11, further comprising: determiningan actual effective visual area of the basic halftone cell based on theactual integer-valued side length of the square zero-shift halftonesupercell; and determining an actual screen frequency based on theactual effective visual area of the basic halftone cell and the printerresolution.
 13. The method of claim 8, further comprising: selecting aneffective visual area of a basic halftone cell of the halftone screen;determining a supercell area based on the selected effective visual areaand the determined integer number of centers.
 14. The method of claim13, further comprising: determining a nominal side length of the squarezero-shift halftone supercell based on the determined supercell area;and determining an actual integer-valued side length of the squarezero-shift halftone supercell based on the determined nominal sidelength of the square zero-shift halftone supercell.
 15. The method ofclaim 14, further comprising determining an actual screen frequencybased on the effective visual area of the basic halftone cell and aresolution of a printer by which the halftone screen will be printed.16. A square zero-shift supercell designing system usable to designing ahalftone screen having a square zero-shift halftone supercell,comprising: a first nominal side length determining circuit, routine orapplication; an actual side length selecting circuit, routine orapplication; and a center number determining circuit, routine orapplication;
 17. The square zero-shift supercell designing system ofclaim 16, further comprising at least one input device usable to inputdata defining a desired angle between a first frame of reference and asecond frame of reference and a desired length for a first side of apair of sides of a non-square supercell in the first frame of reference.18. The square zero-shift supercell designing system of claim 17,wherein the first nominal side length determining circuit, routine orapplication determines a value for a nominal side length for a secondside of the pair of sides of the non-square supercell in the first frameof reference.
 19. The square zero-shift supercell designing system ofclaim 18, further comprising a display device usable to display thedetermined nominal side length for the second side of the pair of sidesof the non-square supercell, wherein the at least one input device isusable to input data defining an actual side length for the second sidesuch that a number of centers within the square zero-shift halftonesupercell is an integer.
 20. The square zero-shift supercell designingsystem of claim 18, wherein the actual side length selecting circuit,routine or application selects an actual side length for the second sidebased on the determined nominal side length of the second side such thata number of centers within the square zero-shift halftone supercell isan integer.
 21. The square zero-shift supercell designing system ofclaim 21, wherein the center number determining circuit, routine orapplication determines an integer number of centers within the squarezero-shift halftone supercell based on the side lengths of the first andsecond sides of the non-square supercell.
 22. The square zero-shiftsupercell designing system of claim 21, further comprising: an effectivevisual area estimating circuit, routine or application; and a supercellarea determining circuit, routine or application; a second nominal sidelength determining circuit, routine or application; an actual sidelength determining circuit, routine or application; an effective visualarea determining circuit, routine or application; and an actual screenfrequency determining circuit, routine or application.
 23. The squarezero-shift supercell designing system of claim 22, wherein the effectivevisual area estimating circuit, routine or application estimates aneffective visual area of a basic halftone cell of the halftone screenbased on a resolution of a printer by which the halftone screen will beprinted and a desired screen frequency.
 24. The square zero-shiftsupercell designing system of claim 23, wherein the supercell areadetermining circuit, routine or application determines a supercell areabased on the estimated effective visual area and the determined integernumber of centers of the square zero-shift halftone supercell.
 25. Thesquare zero-shift supercell designing system of claim 24, wherein thesecond nominal side length determining circuit, routine or applicationdetermines a nominal side length of the square zero-shift halftonesupercell based on the determined supercell area.
 26. The squarezero-shift supercell designing system of claim 25, wherein the actualside length determining circuit, routine or application determines anactual integer-valued side length of the square zero-shift halftonesupercell based on the determined nominal side length of the squarezero-shift halftone supercell.
 27. The square zero-shift supercelldesigning system of claim 26, wherein the effective visual areadetermining circuit, routine or application determines an actualeffective visual area of the basic halftone cell based on the actualinteger-valued side length of the square zero-shift halftone supercell.28 The square zero-shift supercell designing system of claim 27, whereinthe actual screen frequency determining circuit, routine or applicationdetermines an actual screen frequency based on the actual effectivevisual area of the basic halftone cell and the printer resolution. 29.The square zero-shift supercell designing system of claim 21, furthercomprising: a supercell area determining circuit, routine orapplication; a second nominal side length determining circuit, routineor application; an actual side length determining circuit, routine orapplication; an actual screen frequency determining circuit, routine orapplication.
 30. The square zero-shift supercell designing system ofclaim 29, wherein: the at least one input device usable to input datadefining an effective visual area of a basic halftone cell of thehalftone screen; and the supercell area determining circuit, routine orapplication determines a supercell area based on the defined effectivevisual area and the determined integer number of centers.
 31. The squarezero-shift supercell designing system of claim 30, wherein the secondnominal side length determining circuit, routine or applicationdetermines a nominal side length of the square zero-shift halftonesupercell based on the determined supercell area.
 32. The squarezero-shift supercell designing system of claim 31, wherein the actualside length determining circuit, routine or application determines anactual integer-valued side length of the square zero-shift halftonesupercell based on the determined nominal side length of the squarezero-shift halftone supercell.
 33. The square zero-shift supercelldesigning system of claim 32, wherein the actual screen frequencydetermining circuit, routine or application determines an actual screenfrequency based on the effective visual area of the basic halftone celland a resolution of a printer by which the halftone screen will beprinted.
 34. A square zero-shift supercell designing system, comprising:an angle determining circuit, routine or application; a center numberdetermining circuit, routine or application;
 35. The square zero-shiftsupercell designing system of claim 34, further comprising at least oneinput device usable to input data defining a desired length for a firstside and a second side of a pair of sides of a non-square supercell. 36.The square zero-shift supercell designing system of claim 35, whereinthe angle determining circuit, routine or application determines anangle between a first frame of reference and a second frame of referencebased on the input lengths for the first and second sides of the pair ofsides of the non-square supercell.
 37. The square zero-shift supercelldesigning system of claim 35, wherein the center number determiningcircuit, routine or application determines an integer number of centerswithin the square zero-shift halftone supercell based on the sidelengths of the first and second sides of the non-square supercell. 38.The square zero-shift supercell designing system of claim 37, furthercomprising: an effective visual area estimating circuit, routine orapplication; a supercell area determining circuit, routine orapplication; a nominal side length determining circuit, routine orapplication; an actual side length determining circuit, routine orapplication; an effective visual area determining circuit, routine orapplication; and an actual screen frequency determining circuit, routineor application.
 39. The square zero-shift supercell designing system ofclaim 38, wherein the effective visual area estimating circuit, routineor application estimates an effective visual area of a basic halftonecell of the halftone screen based on a resolution of a printer by whichthe halftone screen will be printed and a desired screen frequency. 40.The square zero-shift supercell designing system of claim 39, whereinthe supercell area determining circuit, routine or applicationdetermines a supercell area based on the estimated effective visual areaand the determined integer number of centers of the square zero-shifthalftone supercell.
 41. The square zero-shift supercell designing systemof claim 40, wherein the nominal side length determining circuit,routine or application determines a nominal side length of the squarezero-shift halftone supercell based on the determined supercell area.42. The square zero-shift supercell designing system of claim 41,wherein the actual side length determining circuit, routine orapplication determines an actual integer-valued side length of thesquare zero-shift halftone supercell based on the determined nominalside length of the square zero-shift halftone supercell.
 43. The squarezero-shift supercell designing system of claim 42, wherein the effectivevisual area determining circuit, routine or application determines anactual effective visual area of the basic halftone cell based on theactual integer-valued side length of the square zero-shift halftonesupercell.
 44. The square zero-shift supercell designing system of claim43, wherein the actual screen frequency determining circuit, routine orapplication determines an actual screen frequency based on the actualeffective visual area of the basic halftone cell and the printerresolution.
 45. The square zero-shift supercell designing system ofclaim 37, further comprising: a supercell area determining circuit,routine or application; a second nominal side length determiningcircuit, routine or application; an actual side length determiningcircuit, routine or application; an actual screen frequency determiningcircuit, routine or application.
 46. The square zero-shift supercelldesigning system of claim 45, wherein: the at least one input deviceusable to input data defining an effective visual area of a basichalftone cell of the halftone screen; and the supercell area determiningcircuit, routine or application determines a supercell area based on thedefined effective visual area and the determined integer number ofcenters.
 47. The square zero-shift supercell designing system of claim46, wherein the second nominal side length determining circuit, routineor application determines a nominal side length of the square zero-shifthalftone supercell based on the determined supercell area.
 48. Thesquare zero-shift supercell designing system of claim 47, wherein theactual side length determining circuit, routine or applicationdetermines an actual integer-valued side length of the square zero-shifthalftone supercell based on the determined nominal side length of thesquare zero-shift halftone supercell.
 49. The square zero-shiftsupercell designing system of claim 48, wherein the actual screenfrequency determining circuit, routine or application determines anactual screen frequency based on the effective visual area of the basichalftone cell and a resolution of a printer by which the halftone screenwill be printed.